I was surprised the iPhone YouTube app didn't provide Picture in Picture support by default:
But this video has a lot of views and seems to show a simple way (basically go into settings and enable the feature).
However, when following the video's instructions, the option is not available in settings so it cannot be enabled.
The comments on the video indicate similar experiences.
Note
There is a short exchange in the comments which suggests YouTube Premium may be required, but then suggests that's not the issue (since people who have YT Premium also cannot access the feature):
I’m … trying everything and can’t get it
You need youtube premium.
I have it
Best Answer
This feature used to be in beta on iPhone (as all Google beta features, no one know who will have it).
YouTube recently said in a statement that the feature is launching in the US only for Youtube Premium subscribers first. It will come later for all users. Source
Many Shortcuts provides the PiP mode, you can try one of these : YouTube PiP, PiP JS script